FUZZYMAR

Period: 2019 - 2021

Fuzzy metrics are considered as an appropriate mathematical tool of measurement when such a measurement yields a degree of similarity relative to the value of a parameter. On the other side, indistinguishability operators provide a degree of such a similarity extending the notion of equivalence class to the fuzzy context. In the literature, one can find that the development of both mathematical frameworks is carried out independently and assuming that both notions are unrelated, since fuzzy metrics involve a parameter in its formulation and indistinguishability operators do not. The concept of modular indistinguishability operator, which involves a parameter in its formulation, has been introduced recently. This new kind of operator has allowed unifying under the same framework the aforesaid apparently unrelated fuzzy notions of measuring. This opens a wide range of possibilities which are totally unexplored from both the theoretical and applied viewpoints.

The main theoretical target of the project is to study what properties of fuzzy metrics and indistinguishability operators remain valid in the general framework of modular indistinguishability operators and, thus, try to make clear the differences with the results coming from the two classical frameworks. The extension to the modular context of all topological results for fuzzy metrics and all results of structure and representation for indistinguishability operators will be addressed. Moreover, this viewpoint will help us transfer, to the indistinguishability context, typical results that only could be stated in the fuzzy metric context, and vice versa. Moreover, applications of the new developed results to fixed point theory are expected to be given in such a way that fixed point results in the context of fuzzy metrics will be retrieved as a particular case. Furthermore, relationships between aggregation theory, generalized metric structures and modular indistinguishability operators are also an objective of the theoretical approach of the project.

Regarding the application of the theoretical results, the FUZZYMAR project aims at developing new techniques and algorithms that involve the use of modular indistinguishability operators as similarity measures in: (1) addressing two recurring problems in general robotics, namely robust model fitting and (visual) descriptors comparison and matching, both with also application to computer vision and pattern recognition, and (2) modelling multi-robot systems adopting swarm techniques, to deepen in the analysis of their performance and accordingly devise better coordinated task execution strategies.